# Fibered rotation vector and hypoellipticity for quasiperiodic cocycles in compact Lie groups

## TL;DR

This paper introduces a fibered rotation vector for quasiperiodic cocycles in compact Lie groups and proves that Diophantine conditions on this vector imply smooth reducibility, linking dynamical systems and hypoellipticity.

## Contribution

It defines a fibered rotation vector for quasiperiodic cocycles and establishes a hypoellipticity result under Diophantine conditions, extending the Greenfield-Wallach conjecture.

## Key findings

- Fibered rotation vector is well-defined for almost reducible cocycles.
- Diophantine rotation vectors imply smooth reducibility of cocycles.
- Establishes a connection between dynamical properties and hypoellipticity in PDEs.

## Abstract

Using weak solutions to the conjugation equation, we define a fibered rotation vector for almost reducible quasi-periodic cocycles in $\mathbb{T}^{d} \times G$, $G$ a compact Lie group, over a Diophantine rotation. We then prove that if this rotation vector is Diophantine with respect to the rotation in $\mathbb{T}^{d}$, the cocycle is smoothly reducible, thus establishing a hypoellipticity property in the spirit of the Greenfield-Wallach conjecture in PDEs.